MATHEMATICS TALK Thursday, September 7
OLIN 372 12:00
Student Colloquium Series Presented by:
Karl Voss Department of Mathematics Bucknell University
Title: "Brownian Motion and Stock Prices"
Abstract: Life and many natural phenomena are full of uncertainties. One might wonder, "How can I model something whose behavior is inherently uncertain?" We will consider an elementary introduction to Brownian motion as one way to predict the future behavior of uncertain phenomena. In particular, we will describe a model of stock prices using Brownian motion and look at the implications for some of the current scandals in the financial markets.
MATHEMATICS TALK Thursday, September 21, 2006
OLIN 268 12:00
Student Colloquium Series Presented by:
Linda Smolka Department of Mathematics Bucknell University
Title: "Shocks, Waves, Fans and the Method of Characteristics"
Abstract: First order conservation laws can be used to model many complicated processes, such as, the adsorption of nutrients in the digestive tract, the flow of traffic, or the flow of a thin film on a solid substrate. The unknown quantity in such a law depends on space and time, and the conservation law itself is a partial differential equation, that is, an equation containing partial derivatives in space and time of the unknown function. Using concepts from calculus and ordinary differential equations we show how to derive a traveling wave solution to a first order conservation law using the method of characteristics. We'll also discuss other possible solutions such as, shocks and rarefaction fans, and see an example of a conservation law for thin a film flow.
MATHEMATICS TALK Thursday, October, 2006
OLIN 268 12:00
Student Colloquium Series Presented by:
David Farmer American Institute of Mathematics
Title: "Random numbers"
Abstract: Many phenomena in the real world appear to have an element of randomness. Examples are the distances between cars on a freeway, or the total rainfall in October. Many things in mathematics also appear to be random, even though we know they are not, such as the digits of pi, or the zeros of the Riemann zeta function. I will talk about different types of randomness, and how we use random numbers to study things that are not random.
MATHEMATICS TALK Thursday, October 19, 2006
OLIN 268 12:00
Student Colloquium Series Presented by:
Lisa Clark Department of Mathematics Susquehanna University
Title: "Groupoids and the Battle for the Universe"
Abstract: A groupoid is a small category in which every morphism is invertible. What does this eloquent definition mean and how does one do battle with it? This talk will be geared toward students who have no preconceived ideas or knowledge about groupoids. With a clean slate, I will explain this lovely definition, give some examples, and discuss the appearance of groupoids in our mathematical universe.
Lisa Clark with Ueli Daepp
MATHEMATICS TALK Thursday, November 2, 2006
OLIN 268 12:00
Student Colloquium Series Presented by:
John Meier Mathematics Department Lafayette College
Title: "Embeddability: How Plastic Man Got Me Interested in Topology"
Abstract: One of the most impressive things mathematics can do is prove that certain things cannot be done. One example of this is the utlities graph: Draw three utilities (electric, gas and water) and three houses on a piece of paper, add lines connecting each house to each power plant, and at some point you will have to have two of these lines cross. There is no way to avoid it. In this talk we will talk about such things and their higher dimensional analogs, each of which will be introduced by panels from Plastic Man comics (and a few other comics as well).
John Meier
Seminar Talk - Tuesday - November 9 - 4:00 - OLIN 351
Title: "The Lifting Problem for Commuting Subnormals"
Abstract. We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of Curto, Muhly and Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.
Our tools include the use of 2-variable weighted shifts, the Six-point Test for joint hyponormality, disintegration of measures techniques, the theory of multivariable moment problems, and matrix positivity. We obtain new necessary conditions for the existence of a lifting, and generate new pathology associated with bringing together the Berger measures associated to each individual weighted shift.
For subnormal 2-variable weighted shifts, we then find the precise relationship between the Berger measure of the pair and the Berger measures of the shifts associated to horizontal rows and vertical columns of weights. Finally, we consider the (multivariable) spectral theory of these hyponormal pairs, and discover some unexpected new phenomena, not present in the single variable theory.
Seminar Talk - Thursday - November 9 - 4:00 - OLIN 383Title: "Hyponormality and Subnormality for Toeplitz Operators and Unilateral Weighted Shifts"
Abstract: We discuss the gap between 2-hyponormality and subnormality for Toeplitz operators. In joint work with W.Y. Lee, we have established that every 2-hyponormal Toeplitz operator with either trigonometric symbol or unitarily equivalent to a unilateral weighted shift must be subnormal. On the other hand, in joint work with S.H. Lee and W.Y. Lee we have exploited C. Cowen and J. Long's construction of a nontrivial subnormal Toeplitz operator to provide an example of a nonsubnormal 2-hyponormal Toeplitz operator. We also show that, under a suitable condition on the symbol, a 2-hyponormal Toeplitz operator with nonzero self-commutator must be normal or analytic.
Characterizing subnormality for Toeplitz operators is tied to the search for concrete models for 2-hyponormality. We say that an operator T acting on a Hilbert space H is weakly subnormal if there exists an extension R acting on a bigger Hilbert space K such that T*Tf=TT*f for all f in H; we say that R is a partially normal extension of T. Together with I.B. Jung, W.Y. Lee and S.S. Park we have proved that 2-hyponormal operators are automatically weakly subnormal. More generally, we prove that T is (k+1)-hyponormal if and only if T is weakly subnormal and its minimal partially normal extension is k-hyponormal.
As a direct application, we obtain a matricial representation of the minimal normal extension of a subnormal operator as a block staircase matrix. When this is applied to a weighted shift, we obtain a new, simpler proof of Stampfli's construction of the minimal normal extension of a subnormal weighted shift. Moreover, the above mentioned recursive characterization of k-hyponormality allows us to give a simple proof of C. Gu's description of the gap between k-hyponormality and (k+1)-hyponormality for Toeplitz operators.
MATHEMATICS TALK Thursday, November 16, 2006
OLIN 268 12:00
Student Colloquium Series Presented by:
Annalisa Crannell Department of Mathematics Franklin and Marshall
Title: "Math and Art: The Good, the Bad, and the Pretty"
Abstract: Dust off those old similar triangles, and get ready to put them to new use in looking at art! We're going to explore the mathematics behind perspective paintings---a mathematics that starts off with simple rules, and yet leads into really lovely, really tricky mathematical puzzles. Why do artists use vanishing points? What's the difference between 1-point and 3-point perspective? What's the difference between a perspective artist and a camera? We'll look at all of these questions, and more.
Annalisa Crannell and Paul McGuire
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